What kind of experience do I need to get into an AI master's programme?
Well, mathematics is a very strong field in AI as that is what most AI algorithms revolve around. You only need to learn how to write code (Mostly python or c++, python is easier to learn in my opinion) and how to write mathematical equations in code. Once you have that down it should be enough but I don't how courses work at yours as I know our University require a degree in computer science.

If you want to impress somebody then I can pass you an old University assignment I had about backpropagation in neural networks. It does not involve any code but to determine the partial derivative of an equation that is used to update weights in a neural network. If you do the assignment correctly you should know the basics of how a fully connected neural network works, which is the cornerstone of most neural networks, and also how to train it through backpropagation using only maths. Not only would you know how neural networks work but you would have something to show for it. Witch hopefully could contribute to applying for your Masters. 

Lastly, I know of people that wrote scientific papers in the realm of computer science, to prove they would survive in masters, and then applied for a Master's degree in Computer Science. If you are really serious about it you can do this but just double-check with someone as you might do all the work for naught.